91Ó°ÊÓ

Gender and color preference. A 2001 study asked 1,924 male and 3,666 female undergraduate college students their favorite color. A \(95 \%\) confidence interval for the difference between the proportions of males and females whose favorite color is black ( \(p_{\text {male }}-p\) female \()\) was calculated to be (0.02,0.06) . Based on this information, determine if the following statements are true or false, and explain your reasoning for each statement you identify as false. 2 (a) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) lower to \(6 \%\) higher than the true proportion of females whose favorite color is black. (b) We are \(95 \%\) confident that the true proportion of males whose favorite color is black is \(2 \%\) to \(6 \%\) higher than the true proportion of females whose favorite color is black. (c) \(95 \%\) of random samples will produce \(95 \%\) confidence intervals that include the true difference between the population proportions of males and females whose favorite color is black. (d) We can conclude that there is a significant difference between the proportions of males and females whose favorite color is black and that the difference between the two sample proportions is too large to plausibly be due to chance. (e) The \(95 \%\) confidence interval for \(\left(p_{\text {female }}-p_{\text {male }}\right)\) cannot be calculated with only the information given in this exercise.

Step by step solution

01

Analyze Statement (a)

This statement says that the true proportion of males who prefer black is lower to higher by 2% to 6% compared to females. However, the confidence interval given (0.02, 0.06) supports males having a higher preference. Thus, this statement is incorrect because it reverses the direction of the difference.

02

Analyze Statement (b)

This statement indicates that the true proportion of males who prefer black is between 2% and 6% higher than females, which matches the given confidence interval (0.02, 0.06). Thus, this statement is correct.

03

Analyze Statement (c)

This statement describes a general property of confidence intervals. A 95% confidence interval means that if we were to take many samples and calculate the confidence interval for each sample, 95% of those intervals would contain the actual population difference. This statement is true.

04

Analyze Statement (d)

This statement asserts that the difference in proportions is significant due to the confidence interval not including zero, indicating a likely non-random difference. Hence, this statement is true.

05

Analyze Statement (e)

The confidence interval for (female - male) can be calculated by reversing the signs: it would be (-0.06, -0.02). So, this statement is false because with the given information, the confidence interval for the reverse can be calculated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Proportion Difference

In statistics, understanding the difference between proportions is crucial when comparing two groups. Here, we're looking at the proportion difference between male and female college students who prefer black as their favorite color. This difference is represented by a confidence interval, which in this case is (0.02, 0.06). This range indicates that the proportion of males who prefer black is estimated to be between 2% and 6% higher than the proportion of females.

To put it simply, a proportion difference tells us how one group's preference compares to another group's. Calculating this helpfully quantifies how distinct these groups are in terms of their choices.

When interpreting this, remember that if the interval does not include zero, there is evidence of a meaningful difference. For instance, our interval indicates a positive number range, solidifying that male students indeed have a higher preference for black, making it statistically significant.

Statistical Significance

Statistical significance helps determine if the results of a study are likely due to chance or if they reflect actual differences. In the context of our exercise, the confidence interval (0.02, 0.06) does not include zero, suggesting that the proportion difference is statistically significant.

When the result is statistically significant, it infers that the observed difference between male and female preferences for black cannot be attributed solely to random sampling variability.

• It ensures that the observed effect isn’t likely a fluke.
• This significance is typically evaluated at a specific confidence level, like 95% in this case.

Knowing whether differences are statistically significant informs us about genuine patterns rather than random noise, allowing more confident conclusions about preferences in the population.

Sample Analysis

In sample analysis, data from a subset of a population is used to make inferences about the entire population. In this exercise, data was collected from a substantial number of students—1,924 males and 3,666 females. From these samples, a 95% confidence interval was developed.

Understanding sample analysis is essential because it summarizes whether the sample data accurately reflects the population or if differences are likely due to sampling error.

• A sample must be adequately large and randomly selected to be representative.
• The analysis then estimates population parameters, expressed by confidence intervals.

In our case, the confidence interval tells us how reliable the estimated difference between male and female preferences is and how the sample analysis supports identifying this disparity in color preference.